The accent you suggest as "ancient Greek" is in fact the so-called "Erasmian accent," i.e. a scholarly attempt to reproduce ancient speech devised by Erasmus a.k.a. Desiderius (1469-1536), a Dutch scholar. This invented accent has a very strong latin pitch to it, as some of the Greek letters (eg. theta, delta, gamma) are very hard to pronounced by most western Europeans. Erasmus

What the true accent of classical Athens (let alone Sparta, Crete, Macedonia, etc) was cannot even be a matter of speculation! Think of an English language scholar in 1500 years trying to reconstruct what English sounds like, when there are today diverging accents such as Oxbridge, Cockney, South African, Noo Yawk and New Joysey, Southern, Jive/Ebonic, and DownUnderish!

There is however one universally accepted fact: Greek words that begin with a consonant and have a "rough breathing" (daseia) originally has a slight "h" or "s" hissing to them, mostly lost in modern Greek. This is why the Greek word for water, YDOR, became HYDOR in Latin (as in hyrdoelectric), YPER became HYPER or SUPER, EXIS became SEXUS and so forth.

My personal bias is that it is more correct to read classical texts using use the contemporary Greek pronounciation rather than an accent devised by a well-meaning Dutch scholar. After all, Greek is a living language for several millennia, that has evolved in a pretty linear fashion. It never had to be resurrected from scratch, as in the case of Hebrew. Why? Pretty much for the same reason that we would rather Chaucer today using a contemporary English accent rather than some devised Pythonesque manner.